VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.7%

Time: 8.4s

Alternatives: 10

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	return -(x * (1.0 / tan(B))) + (1.0 / sin(B));
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + (1.0d0 / sin(b))
end function

Java

public static double code(double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
}

Python

def code(B, x):
	return -(x * (1.0 / math.tan(B))) + (1.0 / math.sin(B))

Julia

function code(B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(1.0 / sin(B)))
end

MATLAB

function tmp = code(B, x)
	tmp = -(x * (1.0 / tan(B))) + (1.0 / sin(B));
end

Wolfram

code[B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	return -(x * (1.0 / tan(B))) + (1.0 / sin(B));
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + (1.0d0 / sin(b))
end function

Java

public static double code(double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
}

Python

def code(B, x):
	return -(x * (1.0 / math.tan(B))) + (1.0 / math.sin(B))

Julia

function code(B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(1.0 / sin(B)))
end

MATLAB

function tmp = code(B, x)
	tmp = -(x * (1.0 / tan(B))) + (1.0 / sin(B));
end

Wolfram

code[B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x}{\sin B}, -\cos B, {\sin B}^{-1}\right) \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (fma (/ x (sin B)) (- (cos B)) (pow (sin B) -1.0)))

C

double code(double B, double x) {
	return fma((x / sin(B)), -cos(B), pow(sin(B), -1.0));
}

Julia

function code(B, x)
	return fma(Float64(x / sin(B)), Float64(-cos(B)), (sin(B) ^ -1.0))
end

Wolfram

code[B_, x_] := N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * (-N[Cos[B], $MachinePrecision]) + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}} \]

   2. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} + \frac{1}{\sin B} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \color{blue}{\frac{1}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

   5. un-div-inv N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

   6. lift-tan.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\color{blue}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

   7. tan-quot N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\color{blue}{\frac{\sin B}{\cos B}}}\right)\right) + \frac{1}{\sin B} \]

   8. lift-sin.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\frac{\color{blue}{\sin B}}{\cos B}}\right)\right) + \frac{1}{\sin B} \]

   9. associate-/r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\sin B} \cdot \cos B}\right)\right) + \frac{1}{\sin B} \]

   10. distribute-rgt-neg-in N/A

       \[\leadsto \color{blue}{\frac{x}{\sin B} \cdot \left(\mathsf{neg}\left(\cos B\right)\right)} + \frac{1}{\sin B} \]

   11. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\sin B}, \mathsf{neg}\left(\cos B\right), \frac{1}{\sin B}\right)} \]

   12. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\sin B}}, \mathsf{neg}\left(\cos B\right), \frac{1}{\sin B}\right) \]

   13. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\sin B}, \color{blue}{-\cos B}, \frac{1}{\sin B}\right) \]

   14. lower-cos.f64 99.8

       \[\leadsto \mathsf{fma}\left(\frac{x}{\sin B}, -\color{blue}{\cos B}, \frac{1}{\sin B}\right) \]

   15. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\sin B}, -\cos B, \color{blue}{\frac{1}{\sin B}}\right) \]

   16. inv-pow N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\sin B}, -\cos B, \color{blue}{{\sin B}^{-1}}\right) \]

   17. lower-pow.f64 99.8

       \[\leadsto \mathsf{fma}\left(\frac{x}{\sin B}, -\cos B, \color{blue}{{\sin B}^{-1}}\right) \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\sin B}, -\cos B, {\sin B}^{-1}\right)} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sin B} - \frac{x}{\tan B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (- (/ 1.0 (sin B)) (/ x (tan B))))

C

double code(double B, double x) {
	return (1.0 / sin(B)) - (x / tan(B));
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 / sin(b)) - (x / tan(b))
end function

Java

public static double code(double B, double x) {
	return (1.0 / Math.sin(B)) - (x / Math.tan(B));
}

Python

def code(B, x):
	return (1.0 / math.sin(B)) - (x / math.tan(B))

Julia

function code(B, x)
	return Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)))
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 / sin(B)) - (x / tan(B));
end

Wolfram

code[B_, x_] := N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{1}{\sin B} \]

   2. lift-/.f64 N/A

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{1}{\sin B} \]

   3. un-div-inv N/A

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{1}{\sin B} \]

   4. lower-/.f64 99.8

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{1}{\sin B} \]

4. Applied rewrites 99.8%

   \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{1}{\sin B} \]

5. Final simplification 99.8%

   \[\leadsto \frac{1}{\sin B} - \frac{x}{\tan B} \]
6. Add Preprocessing

Alternative 3: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos B \cdot x}{\sin B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ (- 1.0 (* (cos B) x)) (sin B)))

C

double code(double B, double x) {
	return (1.0 - (cos(B) * x)) / sin(B);
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 - (cos(b) * x)) / sin(b)
end function

Java

public static double code(double B, double x) {
	return (1.0 - (Math.cos(B) * x)) / Math.sin(B);
}

Python

def code(B, x):
	return (1.0 - (math.cos(B) * x)) / math.sin(B)

Julia

function code(B, x)
	return Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B))
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 - (cos(B) * x)) / sin(B);
end

Wolfram

code[B_, x_] := N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}} \]

   2. lift-neg.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} + \frac{1}{\sin B} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \color{blue}{\frac{1}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

   5. un-div-inv N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

   6. lift-tan.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\color{blue}{\tan B}}\right)\right) + \frac{1}{\sin B} \]

   7. tan-quot N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\color{blue}{\frac{\sin B}{\cos B}}}\right)\right) + \frac{1}{\sin B} \]

   8. lift-sin.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\frac{\color{blue}{\sin B}}{\cos B}}\right)\right) + \frac{1}{\sin B} \]

   9. associate-/r/ N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\sin B} \cdot \cos B}\right)\right) + \frac{1}{\sin B} \]

   10. distribute-rgt-neg-in N/A

       \[\leadsto \color{blue}{\frac{x}{\sin B} \cdot \left(\mathsf{neg}\left(\cos B\right)\right)} + \frac{1}{\sin B} \]

   11. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\sin B}, \mathsf{neg}\left(\cos B\right), \frac{1}{\sin B}\right)} \]

   12. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\sin B}}, \mathsf{neg}\left(\cos B\right), \frac{1}{\sin B}\right) \]

   13. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\sin B}, \color{blue}{-\cos B}, \frac{1}{\sin B}\right) \]

   14. lower-cos.f64 99.8

       \[\leadsto \mathsf{fma}\left(\frac{x}{\sin B}, -\color{blue}{\cos B}, \frac{1}{\sin B}\right) \]

   15. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\sin B}, -\cos B, \color{blue}{\frac{1}{\sin B}}\right) \]

   16. inv-pow N/A

       \[\leadsto \mathsf{fma}\left(\frac{x}{\sin B}, -\cos B, \color{blue}{{\sin B}^{-1}}\right) \]

   17. lower-pow.f64 99.8

       \[\leadsto \mathsf{fma}\left(\frac{x}{\sin B}, -\cos B, \color{blue}{{\sin B}^{-1}}\right) \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\sin B}, -\cos B, {\sin B}^{-1}\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{\sin B} \cdot \left(-\cos B\right) + {\sin B}^{-1}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{x}{\sin B} \cdot \left(-\cos B\right) + \color{blue}{{\sin B}^{-1}} \]

   3. inv-pow N/A

      \[\leadsto \frac{x}{\sin B} \cdot \left(-\cos B\right) + \color{blue}{\frac{1}{\sin B}} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{x}{\sin B} \cdot \left(-\cos B\right) + \color{blue}{\frac{1}{\sin B}} \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{\sin B} + \frac{x}{\sin B} \cdot \left(-\cos B\right)} \]

   6. lift-neg.f64 N/A

      \[\leadsto \frac{1}{\sin B} + \frac{x}{\sin B} \cdot \color{blue}{\left(\mathsf{neg}\left(\cos B\right)\right)} \]

   7. distribute-rgt-neg-out N/A

      \[\leadsto \frac{1}{\sin B} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\sin B} \cdot \cos B\right)\right)} \]

   8. unsub-neg N/A

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B} \]

   9. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x}{\sin B} \cdot \cos B \]

   10. lift-/.f64 N/A

       \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

   11. associate-*l/ N/A

       \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

   12. sub-div N/A

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]

   13. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]

   14. lower--.f64 N/A

       \[\leadsto \frac{\color{blue}{1 - x \cdot \cos B}}{\sin B} \]

   15. lower-*.f64 99.7

       \[\leadsto \frac{1 - \color{blue}{x \cdot \cos B}}{\sin B} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]

7. Final simplification 99.7%

   \[\leadsto \frac{1 - \cos B \cdot x}{\sin B} \]
8. Add Preprocessing

Alternative 4: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -1.4) t_0 (if (<= x 0.018) (- (/ 1.0 (sin B)) (/ x B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -1.4) {
		tmp = t_0;
	} else if (x <= 0.018) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / b) - (x / tan(b))
    if (x <= (-1.4d0)) then
        tmp = t_0
    else if (x <= 0.018d0) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -1.4) {
		tmp = t_0;
	} else if (x <= 0.018) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -1.4:
		tmp = t_0
	elif x <= 0.018:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -1.4)
		tmp = t_0;
	elseif (x <= 0.018)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -1.4)
		tmp = t_0;
	elseif (x <= 0.018)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4], t$95$0, If[LessEqual[x, 0.018], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 0.0179999999999999986 < x

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\frac{1}{6} \cdot {B}^{2} + 1}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {B}^{2}, 1\right)}}{B} \]

      4. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{B \cdot B}, 1\right)}{B} \]

      5. lower-*.f64 70.1

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(0.16666666666666666, \color{blue}{B \cdot B}, 1\right)}{B} \]

   5. Applied rewrites 70.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)}{B}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B} - x \cdot \frac{1}{\tan B}} \]

      5. lower--.f64 70.1

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)}{B} - x \cdot \frac{1}{\tan B}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      8. un-div-inv N/A

         \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - \color{blue}{\frac{x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - \color{blue}{\frac{x}{\tan B}} \]

   7. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{\tan B}} \]

   8. Taylor expanded in B around 0

      \[\leadsto \frac{1}{B} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

   10. Applied rewrites 97.9%

       \[\leadsto \frac{1}{B} - \frac{x}{\tan B} \]

   if -1.3999999999999999 < x < 0.0179999999999999986

   1. Initial program 99.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{1}{\sin B} \]
   4. Step-by-step derivation

      1. lower-/.f64 99.4

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{1}{\sin B} \]

   5. Applied rewrites 99.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{1}{\sin B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 0.018:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.102:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot B - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.102)
   (/ (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x) B)
   (- (* 0.16666666666666666 B) (/ x (tan B)))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.102) {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	} else {
		tmp = (0.16666666666666666 * B) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.102)
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	else
		tmp = Float64(Float64(0.16666666666666666 * B) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.102], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(0.16666666666666666 * B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.101999999999999993

   1. Initial program 99.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}}{B} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + 1\right)} - x}{B} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{6} + \frac{1}{3} \cdot x\right) \cdot {B}^{2}} + 1\right) - x}{B} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{3} \cdot x, {B}^{2}, 1\right)} - x}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot x + \frac{1}{6}}, {B}^{2}, 1\right) - x}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3}, x, \frac{1}{6}\right)}, {B}^{2}, 1\right) - x}{B} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{1}{6}\right), \color{blue}{B \cdot B}, 1\right) - x}{B} \]

      9. lower-*.f64 66.9

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), \color{blue}{B \cdot B}, 1\right) - x}{B} \]

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}} \]

   if 0.101999999999999993 < B

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\frac{1}{6} \cdot {B}^{2} + 1}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {B}^{2}, 1\right)}}{B} \]

      4. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{B \cdot B}, 1\right)}{B} \]

      5. lower-*.f64 21.9

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(0.16666666666666666, \color{blue}{B \cdot B}, 1\right)}{B} \]

   5. Applied rewrites 21.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)}{B}} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B} - x \cdot \frac{1}{\tan B}} \]

      5. lower--.f64 21.9

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)}{B} - x \cdot \frac{1}{\tan B}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      8. un-div-inv N/A

         \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - \color{blue}{\frac{x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - \color{blue}{\frac{x}{\tan B}} \]

   7. Applied rewrites 22.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{\tan B}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \frac{1}{6} \cdot \color{blue}{B} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

   10. Applied rewrites 28.4%

       \[\leadsto 0.16666666666666666 \cdot \color{blue}{B} - \frac{x}{\tan B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} - \frac{x}{\tan B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (- (/ 1.0 B) (/ x (tan B))))

C

double code(double B, double x) {
	return (1.0 / B) - (x / tan(B));
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 / b) - (x / tan(b))
end function

Java

public static double code(double B, double x) {
	return (1.0 / B) - (x / Math.tan(B));
}

Python

def code(B, x):
	return (1.0 / B) - (x / math.tan(B))

Julia

function code(B, x)
	return Float64(Float64(1.0 / B) - Float64(x / tan(B)))
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 / B) - (x / tan(B));
end

Wolfram

code[B_, x_] := N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}} \]

   2. +-commutative N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\frac{1}{6} \cdot {B}^{2} + 1}}{B} \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {B}^{2}, 1\right)}}{B} \]

   4. unpow2 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{B \cdot B}, 1\right)}{B} \]

   5. lower-*.f64 61.4

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(0.16666666666666666, \color{blue}{B \cdot B}, 1\right)}{B} \]

5. Applied rewrites 61.4%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)}{B}} \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

   3. lift-neg.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. unsub-neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B} - x \cdot \frac{1}{\tan B}} \]

   5. lower--.f64 61.4

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)}{B} - x \cdot \frac{1}{\tan B}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

   8. un-div-inv N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - \color{blue}{\frac{x}{\tan B}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - \color{blue}{\frac{x}{\tan B}} \]

7. Applied rewrites 61.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{\tan B}} \]

8. Taylor expanded in B around 0

   \[\leadsto \frac{1}{B} - \frac{x}{\tan B} \]
9. Step-by-step derivation

10. Applied rewrites 76.4%

    \[\leadsto \frac{1}{B} - \frac{x}{\tan B} \]
11. Add Preprocessing

Alternative 7: 51.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (- (/ (fma (* B B) 0.16666666666666666 1.0) B) (/ x B)))

C

double code(double B, double x) {
	return (fma((B * B), 0.16666666666666666, 1.0) / B) - (x / B);
}

Julia

function code(B, x)
	return Float64(Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B) - Float64(x / B))
end

Wolfram

code[B_, x_] := N[(N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}} \]

   2. +-commutative N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\frac{1}{6} \cdot {B}^{2} + 1}}{B} \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {B}^{2}, 1\right)}}{B} \]

   4. unpow2 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{B \cdot B}, 1\right)}{B} \]

   5. lower-*.f64 61.4

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(0.16666666666666666, \color{blue}{B \cdot B}, 1\right)}{B} \]

5. Applied rewrites 61.4%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)}{B}} \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

   3. lift-neg.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. unsub-neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)}{B} - x \cdot \frac{1}{\tan B}} \]

   5. lower--.f64 61.4

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)}{B} - x \cdot \frac{1}{\tan B}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

   8. un-div-inv N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - \color{blue}{\frac{x}{\tan B}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)\right)\right)}{B} - \color{blue}{\frac{x}{\tan B}} \]

7. Applied rewrites 61.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{\tan B}} \]

8. Taylor expanded in B around 0

   \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} - \color{blue}{\frac{x}{B}} \]
9. Step-by-step derivation

   1. lower-/.f64 51.8

      \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \color{blue}{\frac{x}{B}} \]

10. Applied rewrites 51.8%

    \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \color{blue}{\frac{x}{B}} \]
11. Add Preprocessing

Alternative 8: 50.2% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -6 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= x -6e-21) t_0 (if (<= x 1.0) (/ 1.0 B) t_0))))

C

double code(double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -6e-21) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x / b
    if (x <= (-6d-21)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -6e-21) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -x / B
	tmp = 0
	if x <= -6e-21:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(-x) / B)
	tmp = 0.0
	if (x <= -6e-21)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = -x / B;
	tmp = 0.0;
	if (x <= -6e-21)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[((-x) / B), $MachinePrecision]}, If[LessEqual[x, -6e-21], t$95$0, If[LessEqual[x, 1.0], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.99999999999999982e-21 or 1 < x

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

      2. lower--.f64 51.3

         \[\leadsto \frac{\color{blue}{1 - x}}{B} \]

   5. Applied rewrites 51.3%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.9%

      \[\leadsto \frac{-x}{B} \]

   if -5.99999999999999982e-21 < x < 1

   1. Initial program 99.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. Add Preprocessing

   4. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \sin B, \tan B\right)}{\sin B \cdot \tan B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{1 + x}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + x}{B}} \]

      2. lower-+.f64 51.5

         \[\leadsto \frac{\color{blue}{1 + x}}{B} \]

   7. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\frac{1 + x}{B}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{B} \]
   9. Step-by-step derivation

   10. Applied rewrites 51.5%

       \[\leadsto \frac{1}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.4% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ (- 1.0 x) B))

C

double code(double B, double x) {
	return (1.0 - x) / B;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 - x) / b
end function

Java

public static double code(double B, double x) {
	return (1.0 - x) / B;
}

Python

def code(B, x):
	return (1.0 - x) / B

Julia

function code(B, x)
	return Float64(Float64(1.0 - x) / B)
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 - x) / B;
end

Wolfram

code[B_, x_] := N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   2. lower--.f64 51.8

      \[\leadsto \frac{\color{blue}{1 - x}}{B} \]

5. Applied rewrites 51.8%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
6. Add Preprocessing

Alternative 10: 27.3% accurate, 19.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ 1.0 B))

C

double code(double B, double x) {
	return 1.0 / B;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 1.0d0 / b
end function

Java

public static double code(double B, double x) {
	return 1.0 / B;
}

Python

def code(B, x):
	return 1.0 / B

Julia

function code(B, x)
	return Float64(1.0 / B)
end

MATLAB

function tmp = code(B, x)
	tmp = 1.0 / B;
end

Wolfram

code[B_, x_] := N[(1.0 / B), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing

4. Applied rewrites 36.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \sin B, \tan B\right)}{\sin B \cdot \tan B}} \]

5. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{1 + x}{B}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + x}{B}} \]

   2. lower-+.f64 24.8

      \[\leadsto \frac{\color{blue}{1 + x}}{B} \]

7. Applied rewrites 24.8%

   \[\leadsto \color{blue}{\frac{1 + x}{B}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{B} \]
9. Step-by-step derivation

10. Applied rewrites 25.4%

    \[\leadsto \frac{1}{B} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))